It's inherently difficult to give a negative answer to a question like this, but here's a technical fact that pushes in that direction:

Let ZFC$_n$ be the subtheory of ZFC gotten by restricting Separation and Replacement to $\Sigma_n$ formulas. By the reflection principle,$^1$ for each $n$ the theory ZFC proves that there is an ordinal $\alpha_n$ such that $V_{\alpha_n}\models$ ZFC$_n$. That is: $$\mbox{For each $n\in\mathbb{N}$, ZFC proves Con(ZFC$_n$).}$$

(Given that ZFC proves the compactness theorem, why doesn't this contradict Godel's incompleteness theorem? Well, the point is that "$\forall n$" and "ZFC proves" don't commute: while ZFC does prove the claim above for each specific $n$, it can't prove the full version.)

We can think of the $V_{\alpha_n}$s as "approximate universes" which behave like universes for all "sufficiently simple" formulas, the point being that if you specify a complexity level ahead of time you can always assume you have an approximate universe appropriate to that complexity level.

Now the compactness theorem now naively suggests that - since we can only ever use finitely many sentences in a given proof - any argument with universes whatsoever can be replaced with one involving just approximate universes, and hence a proof in ZFC. This is of course false, but counterexamples have to be "global" as opposed to "local" - they need to at some point refer to the whole of the universe in question as a single completed object.

For exampe, the way ZFC + universes proves the consistency of ZFC is by showing that a universe $U$ is a model of ZFC. The statement "$U\models$ ZFC" is expressed in the language of set theory by talking about Skolem functions over $U$ (or something morally equivalent), and this takes place in the context of the powerset of $U$. But this sort of thing isn't to my knowledge how universes are applied in algebraic geometry - they instead use a universe to argue that a "sufficiently closed" object exists in that universe, and this "local" argument is exactly the sort of thing that the reflection principle tells us can generally be reduced to ZFC alone.

Basically, a candidate example needs to not just take place inside a universe, but rather over a universe.

That said, there is an obvious place to look for such: arguments using two (or $n$) universes. The larger universe does see the smaller universe as a completed object, so the coarse heuristic above suggests that we can replace only the larger universe with an approximate universe - that is, that arguments which are quickly phrased in terms of two universes can be directly translated to arguments involving only one universe. Now we can't cheat anymore - we nee actual arguments about algebraic geometry. My understanding is that we're still in a situation where universes are an unnecessary convenience, but now I'm far outside of my area of competence. Still, the above should give an indication of why a real essential use of universes in a concrete result (which will certainly only involve reference to a small fragment of the cumulative hierarchy) would be very surprising.

$^1$OK fine, the reflection principle is usually phrased for finite subtheories of ZFC. But $(i)$ that's not really any different as far as the heuristic is concerned, just more annoying; and $(ii)$ the stronger version of reflection I've stated is also true (the point being that for each $n$, the schemes of $\Sigma_n$-Separation and -Replacement can be expressed in the language of set theory by a single sentence, which in turn can be proved from finitely many of the ZFC axioms).